Mathematics and Model Rockets

Mathematics and Model Rockets A Teacher s Guide and Curriculum for Grades 5-12 Developed by Sylvia Nolte, Ed. D. Based on coursework by Harold McConnell, Ph. D. Edited by James H. Kranich Jr., P.E. and Ann Grimm EstesEducator.com educator@estesrockets.com 800.820.0202 2012 Estes-Cox Corp.

ESTES 2 EDUCATOR

TABLE OF CONTENTS Introduction...4 Goals and Outcomes...4-5 General Background for the Teacher...5-7 Unit Plans...8-45 1. Lesson 1: Introduction to Unit Demonstration: Launching a Model Rocket...8-10 2. Lesson 2: Finding the Center of Mass...11-13 3. Lesson 3: Finding the Center of Pressure...14-19 4. Lesson 4: Rocket Stability...20-23 5. Lesson 5: Math and Rocket Flight...24-41 Launching a Rocket Gathering Data and Statistics Calculations 6. Lesson 6: (Extension) Aerial Photography...42-44 7. Lesson 7: (Extension) Launching Payloads...45-46 Bibliography...47 Appendix Blackline Masters, Student Worksheets...48-84 Overhead Transparencies...85-97 ESTES 3 EDUCATOR

INTRODUCTION MATHEMATICS AND MODEL ROCKETS GOALS Model rocketry is an extremely useful tool for teaching students in a math classroom. Model rocketry captures the students interest and involves them in applying math concepts in a real and authentic way. It involves them in experimenting and testing their ideas. Using model rocket projects in a math curriculum aids students in learning to use creative and critical thinking and problem-solving skills. It provides opportunities for them to discover underlying principles of mathematics. Using rocketry also allows curriculum integration by combining math, science, craftsmanship, physical education, prediction, language arts, history, technology and research methods. It provides opportunities for cooperative learning as well as experiences that will have definite personal meaning to the students through the constructing and launching of their own rocket. Two other curriculum guides from Estes that may be useful are, Science and Model Rockets for Grades 5, 6, 7, 8 and Physics and Model Rockets for Grades 8, 9, 10 & 11. Science and Model Rockets has units that give directions for constructing model rockets, doing simple altitude tracking, simple math and data collection. Physics and Model Rockets relates Newton s Laws of Motion to model rocketry. These two guides can be used with the current guide to develop a well-integrated science and math unit. This curriculum guide is constructed so that teachers of students from fifth through twelfth grade will find it useful. Units include activities that can be used to support basic concepts and to extend students who have the knowledge and ability to work at a more advanced level. The guide can be used as a complete unit or individual projects may be selected to use if the students have the math and science background needed to understand it. Each group of students is different with a variety of experience and interests. Providing a number of projects allows the teacher and the students to work on activities related to the design of model rockets and the mathematical analysis of the effects of the design on model function that relate to those individual experiences and interests. The teacher will be able to adapt the unit to fit an individual class. Students will understand the connection between math and science. Students will use math in practical applications related to rocketry. Students will recognize the importance of careful construction of a model rocket for stability, safety and a successful launch. ESTES 4 EDUCATOR

STUDENT OUTCOMES THINKING SKILLS The student will be able to: Determine the center of mass experimentally, graphically and/or mathematically. Recognize the importance of the center of mass in rocket flight as the point around which an unstable rocket tumbles. Recognize the significance of the location of the center of pressure in relation to the location of the center of gravity(center of mass) in providing stability for a model rocket. Determine mathematically the centroid and area of a variety of shapes. Determine the lateral center of pressure for a model rocket. Recognize the importance of stability in model rocket flight and test the stability of their own rocket. Track the flight of model rockets to gather data to calculate height, velocities and accelerations. Use mathematical equations to determine velocities and accelerations during the rocket flight and describe their relationship to Newton s Second Law. Observing Reading and following directions and diagrams Problem-solving Analysis GENERAL BACKGROUND FOR THE TEACHER Stability is the single most important consideration in designing a model rocket. The more stable the rocket, the more it tends to rotate (weathercock) into the wind during flight. A stable rocket is one that flies in a smooth, uniform direction. An unstable rocket flies along an erratic path, sometimes tumbling or changing direction. Unstable rockets are dangerous because it is not possible to predict where they will go. All matter, regardless of size, mass or shape, has a point inside called the center of mass. The center of mass is the exact spot where all of the mass of that object is perfectly balanced. The center of mass of an object such as a ruler can be demonstrated by balancing it on your finger. The center of mass can be demonstrated graphically to determine the effect of adding increasing weight to one end of an object such as a rocket body tube. In addition to the center of mass, the center of pressure inside the rocket affects its flight. When air is flowing past a moving rocket it rubs and pushes against the outer surface of the rocket, causing it to begin moving around the center of mass. Think of a weather vane. The arrow is attached to a vertical rod that acts as a pivot point. The arrow is balanced so that the center of mass is right at the pivot point. When the wind blows, the ESTES 5 EDUCATOR

arrow turns, and the head of the arrow points into the oncoming wind. The tail of the arrow points in the downwind direction. The reasons for this is that the tail of the arrow has a much larger surface area than the arrowhead. The flowing air imparts a greater force to the tail than the head and the tail is pushed away. Similar to the concept of the center of mass, there is a precise point in the arrow where all the aerodynamic forces acting on it are perfectly balanced. This spot is called the center of pressure. It is not the same as the center of mass. The center of pressure is between the center of mass and the tail end of the arrow. The tail end has more surface area than the head end. The lateral center of pressure has to do only with the forces applied to the surface directly by air currents. The larger the surface the greater the forces will be. Stability is dependent upon the relationship between the center of mass and the center of pressure. The center of pressure in a rocket must be located toward the tail and the center of mass must be located toward the nose. If they are in the same place or very near each other, then the rocket will be unstable in flight. The center of mass may be moved forward by adding weight to the nose of the rocket. The center of pressure may be moved toward the rear by moving the fins back, increasing their size or by adding fins. The center of pressure can be moved forward by using smaller fins. Calculating the ratio of length to diameter in a rocket will help determine the potential for stability in a design. An ideal ratio is 10 to 1 (a 1-inch diameter rocket with a length of 10 inches, for example). Newton s Three Laws of Motion 1. Objects at rest will stay at rest, and objects in motion will stay in motion in a straight line at constant velocity unless acted upon by an unbalanced force. To understand this law it is necessary to understand the terms: rest, motion and unbalanced force. Rest and motion can be thought of as opposite. Rest is the state of an object when it is not changing position in relation to its surroundings. Rest cannot be defined as a total absence of motion because it could not exist in nature. All matter in the universe is moving all the time, but in the first law of motion, motion means changing position in relation to surroundings. When an object is at rest, the forces acting upon it are balanced. In order for an object to begin moving, the forces acting upon it must become unbalanced. A model rocket is at rest when it is on the launch pad. The forces acting upon it are balanced. The force of gravity is pulling the rocket downward and the rocket launch pad is pushing against it holding it up. When the propellant in the engine is ignited, that provides an unbalanced force. The rocket is then set in motion and would stay in a straight line until other unbalanced forces act upon it. ESTES 6 EDUCATOR

2. Force is equal to mass times acceleration. This is really a mathematical equation, f = ma. This equation applies to launching the rocket off the launch pad. It is essential to understand that there are four basic forces operating on any object moving through the air. These are lift, drag, gravity and thrust. In the context of this unit, the concept of thrust will be emphasized. Thrust is a forward propulsive force that moves an object. In a model rocket, thrust is produced by the rocket engines. Thrust must be greater than the weight of the rocket in order to overcome gravity and lift off from the earth. As the engine ignites and thrust develops, the forces become unbalanced. The rocket then accelerates skyward with the velocity increasing from its initial state at rest (velocity = 0). When examining how thrust is developed in a rocket engine, force in the equation can be thought of as the thrust of the rocket. Mass in the equation is the amount of rocket fuel being burned and converted into gas that expands and then escapes from the rocket. Acceleration is the rate at which the gas escapes. The gas inside the rocket does not really move. The gas inside the engine picks up speed or velocity as it leaves the engine. The greater the mass of rocket fuel burned and the faster the gas produced can escape the engine, the greater the thrust of the rocket. 3. For every action there is always an opposite and equal reaction. A rocket can lift off from a launch pad only when it expels gas out of its engine. The rocket pushes on the gas and the gas pushes on the rocket. With rockets, the action is the expelling of gas out of the engine. The reaction is the movement of the rocket in the opposite direction. To enable a rocket to lift off from the launch pad, the action or thrust from the engine must be greater than the weight of the rocket. NOTES ESTES 7 EDUCATOR

UNIT PLAN Lesson 1 - Introduction (1 or 2 days) Demonstration: Launching a Model Rocket Objectives of the Demonstration: The student will be able to: Generate individual questions and predictions regarding model rockets and the launch. Recognize the need for following the NAR Model Rocketry Safety Code and point out the use of the safety code during the demonstration launch. Participate in the demonstration launch as an observer or as a recovery crew member. Identify and describe the stages of a model rocket flight. Complete an interest survey based on individual knowledge of rockets and plans for the individual construction and testing of a model rocket. Discuss the course goals and objectives including the value of building and launching a model rocket to test predictions and what they will need to know to build and launch a model successfully. BACKGROUND FOR THE TEACHER Launching a model rocket at the beginning of the unit will capture the interest of the students and lead them to consider questions they may have about how and why rockets function as they do. Launching a rocket and evaluating the flight of the rocket provides a set for the unit and also provides motivation for the students to build their own model. STRATEGY Materials needed for each student: A copy of the NAR Safety Code, a copy of the Focus sheet and a copy of the interest survey. Students should have a manila envelope or a folder for the materials and sheets that will be accumulated during this unit. Books, pamphlets, design sheets and catalogs related to model rocketry. Materials needed for the demonstration launch: Your Estes supplier can provide you with the information you need to acquire an electrical launch system. Recommended launch area: ENGINES SITE DIAMETER MAXIMUM ALTITUDE Feet/Meters Feet/Meters 1/2 A 50/15 200/61 A 100/30 400/122 B 200/61 800/244 C 400/122 1600/488 D 500/152 1800/549. Minimum launch site in dimension for a circular area is diameter in feet/meters and for rectangular area is shortest side in feet/meters. Choose a large field away from power lines, tall trees and low-flying aircraft. The larger the launch area, the better your chance of recovering the rocket. The chart above shows the smallest recommended launch areas. Football fields, parks and playgrounds are good areas. Make sure the launch area is free of obstructions, dry weeds, brown grass or highly flammable materials. ESTES 8 EDUCATOR

A. Before the demonstration launch, distribute the focus sheet or display a transparency on the overhead projector. (Appendix ) Ask the students to respond to the questions in the first two columns. After the launch, discuss the questions in the remaining columns. Ask students to fill in the remaining columns. This technique builds support for the unit because it demonstrates value for what the students know and what they want to know. What do I know about model rockets? What do I want to learn about model rockets? After the launch: What is a new idea or knowledge that I have since watching the launch? B. Distribute a copy of the NAR model rocket safety code to each student and review the safety code by using the demonstration rocket and going over each step of the demonstration launch and flight. C. Appoint several people to be the recovery crew - people who follow the flight, recover and return the rocket to the launch pad. D. Using the Overhead Projector, briefly review the flight sequence of a model rocket. (Appendix A) Launch the rocket following the directions...... Remind the students to watch for the following: Steps of model rocket flight sequence and the performance of the rocket. Option: Each student could be given a copy of the flight sequence to record their observations. E. After the rocket flight, process the sequence with the students and discuss their evaluation of the performance of the rocket at each stage. F. An interest survey may also be used. (Appendix ) The interest survey can be useful for the student and for the teacher. It will help the students focus on their own interests and questions so that the project can be meaningful to them individually. It can help guide the teacher s focus within the parameters of the objectives and goals for the course. It is important to provide a number of books, pamphlets, catalogs and design sheets to give students ideas for building their model rockets and ideas for the project types and for the study projects they want to do. Interest Survey ESTES 9 EDUCATOR After the launch: What did I see that I would like to know more about? Name 1. Have you ever built a model rocket? Y N How many have you built? 2. List three things that were positive about the model rockets you have built. List three problems you had with building a model rocket. 3. List at least three things you would like to know more about having to do with space and rocketry. As you make a choice of study projects, think about the things you would specifically like to know so that your project fits your interests. 4. You will participate in several study projects with your model rocket.: a. Altitude tracking b. Data reduction c. Speed of a model rocket d. Photography from a model rocket (optional) e. Egg lofting rocket (optional)

G. Discuss with the students the value of building and launching their own model rockets to test out their ideas. Using their observations of the flight sequence, discuss with the students what they will need to know in order to build and launch a model successfully. Evaluation: Observation of student participation and questions. Review student work in notebook on focus sheet, survey and flight sequence observations. NOTES ESTES 10 EDUCATOR

Lesson 2 - (1 or 2 days) Finding the Center of Mass Objectives of the Lesson: The student will be able to: Find the center of mass of an object experimentally. Change the center of mass of a rocket body tube and graph the findings. Recognize the role of center of mass in rocket stability. BACKGROUND FOR THE TEACHER The center of mass (center of gravity) is the exact spot where all of the mass of that object is perfectly balanced. One of the ways that the center of mass can be found is by balancing the object, such as a ruler, on your finger. If the material used to make the ruler is of uniform thickness and density, the center of mass should be at the halfway point between one end of the stick and the other. If a nail were driven into one of the ends of the ruler, then the center of mass would no longer be in the middle. The balance point would be nearer the end with the nail. The center of mass of an object can be changed by adding weight to one part of the object. For students who are building model rockets, an important concept is the continuing effect on the center of mass that occurs as more weight is added. The center of mass moves less distance as more weight is added. Students can graph this effect. VOCABULARY Center of mass: The point at which the mass of an object such as a model rocket is evenly balanced. Center of gravity: The point in a rocket around which its weight is evenly balanced; the point at which a model rocket will balance on a knife edge; the point at which the mass of the rocket seems to be centered. Abbreviation: CG. Symbol: STRATEGY Materials needed for each student: Rocket body tubes. A few wooden dowel sticks about two feet long and small weights that can be attached safely to the sticks (three lead weights weighing about one gram each for each student or group of students). NOTE - Each student will need the supplies necessary to construct a model rocket for this unit. Estes rocket kits can be purchased individually or Bulk Packs are available which contain the components for twelve model rockets. Teachers can allow class time for students to build a rocket or the students can work on these outside class time. ESTES 11 EDUCATOR

MOTIVATION: Briefly discuss with the students the concept of the center of mass and its importance in model rocket flight. Wrap a piece of colored tape around the mid-point of a wooden dowel then toss it vertically and ask the students to observe what the stick does. Ask the students to describe what they observed. Toss it again and ask them to think about the concept of center of mass. Distribute dowel sticks (each with a midpoint mark) to groups of students. Ask them to toss the stick vertically and horizontally, hard and easy. They should begin to make generalizations about the fact that the stick always rotates around its center. Attach a weight to one end of each stick. Ask the students to observe what changes with the added weight. This time the point about which it rotates will be closer to the weighted end. Show the students that by taking the weighted stick and balancing it across a sharp edge you will find that the new center of gravity has shifted towards the weight and that this is the point about which it now rotates when tossed in the air. Wrap colored tape at this new point and observe again as the dowel is tossed. A. Discuss with the students how this experiment relates to model rocket flight. Points to be made: The experiment shows how a free body in space rotates around its center of gravity. A model rocket in flight is a free body in space. If, for any reason, a force is applied to the flying rocket to cause it to rotate, it will always do so about its center of gravity. Rotating forces applied to rockets in flight can result from lateral winds, air drag on nose cones, weights off-center, air drag on launch lugs, crooked fins, engine mounted off-center or at an angle, unbalanced drag on fins or unequal streamlining. Since rotating forces will always be present, your rocket must be designed to overcome them. If not, it will loop around and go everywhere and end up going nowhere. Discuss the demonstration launch in relation to the stability of the rocket. B. In the following activity the students discover graphically that the movement of the center of mass becomes smaller as more mass is added. This concept will be useful as students begin to understand the importance of the relationship between the center of mass (center of gravity) and the center of pressure in rocket stability. All age students - elementary students will require more guidance. Materials needed for each student: A rocket body tube, three lead weights about one gram in mass, graph paper with 1 cm. squares or worksheet in the appendix and a centimeter ruler. ESTES 12 EDUCATOR

Distribute a rocket body tube to each student. Review with the students the concept of center of mass. Discuss how the students could determine the center of mass of the rocket body tube. Each student should find the center of mass by balancing the body tube on their finger. Students should mark the point where it balances with a colored pencil. They can use the symbol for center of mass: Students can work in pairs or small groups, but each student should have a rocket body tube and should graph their own findings. Elementary students may need assistance and support as they participate in the graphing activity. The graph (Appendix ) is appropriate for elementary and middle school students. Older students can use graph paper. Let each student weigh their own weights to make certain they know the weight. Directions: 1. Place your first weight in one end of the body tube. Find the new center of mass by rebalancing the body tube across your finger. Mark the new center of mass with a different color pencil. Repeat the procedure by adding one weight at a time. Mark each center of mass on the body tube. 2. Measure the distance between the ends of the rocket tube and the first center of mass you determined without weights and plot that on the graph. Measure the distance the center of mass moved with one weight and plot that on your graph. Continue until you have plotted all the changes. 3. Discuss the following questions: What can you observe about the movement of the center of mass as more weight or mass is added? Which way does the center of mass move? Does it move toward or away from the added weights? Was the movement smaller or larger? Why do you think this happens? Evaluation: Observation of student participation, review student work in notebook on results of graphing. NOTES ESTES 13 EDUCATOR

Lesson 3 (2-4 days) Finding the Center of Pressure Objectives of the Lesson: The student will be able to: Recognize the role of center of pressure in rocket stability. Determine the approximate center of pressure of geometric shapes experimentally. Determine the approximate center of pressure of a model rocket using a cutout of the model. Calculate the lateral center of pressure of a model rocket using the math formulas for determining the centroid of a shape and the area of a shape, including a rectangle, a semi-ellipse, a parabola, a triangle, a semi-circle and an ogive to use in calculation of the lateral center of pressure for the entire rocket. BACKGROUND FOR THE TEACHER The center of pressure is the point where all the aerodynamic forces are balanced. Aerodynamic forces forward of this point are equal to the aerodynamic forces behind this point. If a model rocket is suspended at its center of pressure in a moving stream of air, the rocket will not attempt to return to forward flight should it become aimed away from the forward orientation. The lateral center of pressure has to do only with the forces applied to a surface directly by air currents. The larger the surface the greater the forces will be. To guide itself in a forward direction while moving through the air, the rocket must have its center of gravity forward of its center of pressure. For model rockets, the distance between the center of gravity and the center of pressure should be at least one caliber (the diameter of the body tube). Generally, the greater this distance, the more stable the rocket. If the separation of these two points is less than this distance, the rocket may be unstable. Unstable rockets will not return to their original flight path if disturbed from this path while in motion. Stability in a model rocket can be defined as the tendency of a rocket to travel in a straight course with the direction of its thrust despite rotating forces caused by outside disturbances. A stable rocket will travel in a relatively straight upward direction, without tumbling and with minimum oscillation. In a later unit, students will test the stability of their own model rocket through the use of a swing test or the use of a wind tunnel. ESTES 14 EDUCATOR

UNSTABLE ROCKET STABLE ROCKET Younger students can be helped to understand the center of pressure by finding the balance point of a number of geographic shapes used in the construction of model rockets. (See Appendix ) This balancing point, referred to as the centroid, describes the geometric center of area for a given shape. The term x locates the centroid from a reference point. EQUATIONS: Semi-circle x = 4r 3π A = πr2 2 x Triangle x = h 3 A = bh 2 x Rectangle or Square x = a 2 A = ab x ESTES 15 EDUCATOR

Semi-ellipse x = 4a 3π A = πab 2 Parabola x = 2a 5 A = 4ab 3 Ogive x.38a A = r 2 ( 2θ sin2θ ) 2 θ in radians 360 = 2π radians a = r sin θ Calculating the center of pressure of rocket outline x = ΣAx ΣA = A1x1 + A2x 2 + A3x 3 + A4x 4 A1 + A2 + A3 + A4 A = Area of each section x = distance from the centroid of each section to the base. VOCABULARY Center of pressure: The center for all external aerodynamic forces on the complete rocket including the body and fins. Abbreviation: CP Symbol: Centroid of a shape: Determined through mathematical calculation or experimentation to provide data for determining the center of pressure. The centroid of a twodimensional shape defines the geometric center of its area. ESTES 16 EDUCATOR

STRATEGY Materials needed for each student: Packet containing geometric shapes, worksheet for determining centroids and area of each shape mathematically, outline of rocket shape and equation for determining lateral center of pressure. Each student will need a centimeter ruler for measuring shapes and rocket outline. For advanced math students, sheets containing center of pressure equations for a variety of rocket configurations. MOTIVATION: The major concept to achieve in this unit is that the center of pressure is related to the area of a rocket. Use a low friction pivot to demonstrate lateral center of pressure or the overhead transparency demonstrating the center of pressure (OH - Lesson 3). A (A low friction pivot consists of two needle points held rigidly in place on opposite sides of the Airflow object by a heavy wire or board frame-work. The needle points are placed against the object just tightly enough to hold it, without inter- B fering with its rotating on the axis C Figure 4 created between the two points.) Demonstration: Place a two foot long piece of dowel on a low friction pivot as shown in A of Figure 4. Hold the dowel in a uniform air current of ten to fifteen miles per hour. If the pivot has been placed in the center of the dowel and if the dowel is uniform in size (area), the forces exerted by the air pressure will be equal on both sides of the pivot and the air current will produce no rotating effect. If a low friction pivot is not available, the same thinking process can be achieved by using the overhead transparency as a basis for discussion. Ask the students to observe the effect of the wind on the dowel in this position and allow them to guess why there is no rotating effect. In this condition, the center of gravity and the center of pressure will be at the same point. Add a vane of three inch by three inch cardboard by gluing it to one end of the dowel. Put it in the air stream with the pivot in the same position. Ask the students to observe the effect of the air current on the dowel and allow them to guess why the effect is different. The moving air current will exert the greatest force against the end of the dowel which has the vane attached to it. This will cause the dowel to rotate until the end away from the vane points into the wind. Ask the students to make predictions about where the pivot could be moved so that the rotating effect is stopped. The pivot should be moved closer to the vane end of the dowel until a point is located where equal air pressure will be applied to both ends. The air current will no longer cause any part of the dowel to point into the wind. This point is called the lateral center of pressure. The lateral center of pressure has to do only with the forces applied to the surface directly by air currents and the larger the surface the greater the forces will be. ESTES 17 EDUCATOR

Aerodynamic forces acting on a rocket are like a teeter-totter (fulcrum and lever). Not only does the magnitude of the force affect the balance, but also how far away from the pivot point the force acts. The smaller force w acting at a distance d 2 has the same effect of a larger force W acting at a distance d 1. This is just like the aerodynamic forces acting on a rocket. Forces acting at a distance away from the pivot point (or CG for a rocket) create a Moment ; force times distance. In the example above, the moments created by W and w are balanced. A. Distribute packets. Discuss that each one of these shapes has a center of pressure which can be approximately determined by finding the balance point of each one. Point out that these are shapes commonly found in model rockets, the nose cone, the body and the fins. The centroid of these shapes can also be determined mathematically and is a factor in determining the lateral center of pressure of a model rocket. Connect the experiment with the dowel and the cardboard vane to the concept of center of pressure. For all students from elementary through high school: Students may work in pairs or in small groups but each student should have a set of shapes. The shapes may be cut out and used as is if a heavy bond paper is used when reproducing the shapes. If the teacher prefers, students may trace the shapes onto tag board and then cut them out. Using a colored pencil, each student should mark on the shape of the square the point where they think the center of pressure is. Next, ask them to balance the square on a finger and mark the point of the actual center of pressure (the point where the shape balances on a finger). Continue the same process with each shape. For some elementary students and middle school students with guidance and for high school students: Students should complete the packet for Lesson 3, which includes equations for determining the centroids and areas of the geometric shapes and model rockets. (Appendix) Elementary students may need to be guided through the equations step-by-step. Middle school and high school students can use the equations after some demonstration or review. The computer program, Estes ASTROCAD : Performance Analysis for Model Rocketry, allows students to check their calculations or to determine the center of pressure given measurements without going through the equations. However, both methods help students thoroughly understand the mathematics involved. For younger students or those without sufficient math background, a quick way to determine the approximate CP of a model rocket is to make an exact-size cutout of the model (a profile) in cardboard. The point where the cutout will balance on the edge of a ruler will be the approximate lateral center of pressure. Air pressure applied to a surface is proportional to the area of the surface, so the cutout allows the student to approximate the rotating effect of the action of the air pressure. ESTES 18 EDUCATOR

Distribute a sheet of cardboard to each student. Students should make a cutout of their own rocket. Lay the rocket over the piece of cardboard and mark around the edges. Cut around the lines and balance the cutout on a knife edge or ruler edge. Mark the center of pressure of the cutout. These methods determine the lateral center of pressure (the center of pressure with the air currents hitting the rocket broadside). If the rocket is designed so the lateral center of pressure is one body diameter behind the center of gravity it will have ample stability under all reasonable conditions. If, however, the rocket s fins are very crooked, set at opposing angles or if the rocket uses a disc or cone for stabilizing, the lateral center of pressure should be set at least 1 1/2 diameters behind the center of gravity. A more accurate method for calculating the center of pressure is available in the Appendix, Lesson 7. This method applies the same concept, but uses much more detailed information. Extension Using the model outline, students may redraw the fins making larger triangles or making the fins square or rectangular. Using the equations, they can recalculate the center of pressure. NOTES ESTES 19 EDUCATOR

Lesson 4 (2-3 days) Rocket Stability Objectives of the Lesson: The student will be able to: Describe the relationship between the location of the center of mass and the center of pressure in the stability of a model rocket. Discuss the importance of stability in model rocket flight. Test the stability of own model rocket using either the swing test or the wind tunnel test. Correct the design and construction of own model rocket to provide stability. For this unit, the student needs to have constructed a model rocket with fins attached and the engine mounted. The launch lug needs to be attached as well. BACKGROUND FOR THE TEACHER Stability is dependent upon the relationship between the center of mass and the center of pressure. The center of pressure in a rocket must be located toward the tail and the center of mass must be located toward the nose. If they are in the same place or very near each other, the rocket will be unstable in flight. As the students have seen, the center of mass may be moved forward by adding weight to the nose of the rocket. The center of pressure may be moved toward the rear by moving the fins back, increasing their size or by adding fins. The center of pressure can be moved forward by using smaller fins. In flight, the rocket will not be traveling sideward, but with its nose pointed into the wind. With the model s nose pointed into the wind, the location of the effective center of pressure will be affected by the shape of the fins, the thickness of the fins, the shape of the nose cone and location of the launch lug. With most designs this shift is to the rear, adding to the stability of the rocket. It is important to emphasize to the students how essential it is to construct the rocket carefully and for the need to have a rocket that is stable. If a model rocket starts to rotate in flight, it will rotate around its center of gravity. When it turns, the air wind rushing past it wind will then hit the rocket at an angle. If the center of pressure is behind the center of gravity on the model, the air d pressure will moment d exert the greatest force against moment the STABLE UNSTABLE ESTES 20 EDUCATOR

fins. This will counteract the rotating forces and the model will continue to fly straight. If the center of pressure is ahead of the center of gravity, the air currents will exert a greater force against the nose end of the rocket. This will cause it to rotate even farther. Once it has begun rotating it will become unstable. Remember, the lateral center of pressure is the one point where the aerodynamics forces act. In the illustrations, all the wind can then be treated as a single force acting through the center of pressure. This force acting at a distance away (d) from the CG creates a moment that either stabilizes or destabilizes the rocket. It is best to build a rocket with its fins as far as possible to the rear. The farther behind the center of gravity the center of pressure is placed, the stronger and more precise will be the restoring forces on the model and it will fly straighter with less wobbling and side-to-side motion, which robs the rocket of energy. Fins usually should not be placed forward of the center of gravity on a model because this will add to instability. If fins are added forward of the center of gravity, be certain the center of pressure remains behind the center of gravity. Students can also test the stability of a rocket with precision experimentally through the use of the swing test or the use of a wind tunnel.(directions for building a wind tunnel may be found in Estes The Classic Collection, Technical Report TR-5, Building a Wind Tunnel ). The simplest, least expensive method is the swing test. The rocket to be tested should have its engine in place as it would be in flight. With the engine installed, the center of gravity is shifted further to the rear placing the center of gravity at its most critical position. The rocket is suspended from a string. The string is attached around the rocket body using a loop. Slide the loop to the proper position so the rocket is balanced, hanging perpendicular to the string. Apply a small piece of tape to hold the string in place. If the rocket s center of gravity falls in the fin area, it may be balanced by hooking the string diagonally around the fins and body tube. A straight pin may be necessary at the forward edge of one of the fins to hold the string in place. This string mounting system provides a low friction pivot about which the rocket can rotate freely. If a wind tunnel is being used, slide a soda straw along the string to a position just above the rocket. Then suspend the rocket in a low velocity air stream with the nose of the rocket pointing into the wind. Then turn the rocket approximately 10 out of the wind to see if it recovers. If so, the rocket is stable enough for flight. The swing test method involves swinging the suspended rocket overhead in a circular path around the individual. If the rocket is stable, it will point forward into the wind created by its own motion. If the center of pressure is extremely close to the center of gravity, the rocket will not point itself into the wind unless it is pointing directly forward at the time the circular motion is started. This is accomplished by holding the rocket in one hand, with the arm extended, and then pivoting the entire body as the rocket is started in the circular path. Sometimes several tries are needed in order to achieve a perfect start. If it is necessary to hold the rocket to start it, additional checks should be made to determine if the rocket is flight-worthy. Small wind gusts or engine misalignment can cause a rocket that checks out stable when started by hand as described above to be unstable in flight. To be sure that the rocket s stability is sufficient to overcome these problems, the rocket is swung overhead in a state of slight imbalance. Experiments indicate that a single engine rocket will have adequate stability for a safe flight if it remains stable when the above test is made with the rocket rebalanced so the nose drops below the tail with the rocket body at angle of 10 from the horizontal. ESTES 21 EDUCATOR

With cluster powered rockets a greater degree of stability is needed since the engines are mounted off center. The cluster powered rocket should be stable when unbalanced to hang at 15 from the horizontal. Heavier rockets which accelerate at a lower rate require a similar margin of stability. Caution should be exercised when swinging rockets overhead to avoid a collision with objects or people nearby. It is possible to achieve velocities of over 100 miles per hour. This can cause injury. If the student has constructed a rocket that will not be stable, do not attempt to fly it. Corrections have to be made. Lack of stability will cause the rocket to loop around and around in the air. It will seldom reach over 30 feet in height and can never reach a velocity of more than 20 or 30 miles an hour. Also, it is possible that a rocket could suddenly become stable after making a couple of loops, due to the lessening of the fuel load. This could cause it to fly straight into the ground and could cause serious injury or damage. If a rocket does not show the degree of stability required for safety it can be easily altered by either moving the center of gravity forward or by moving the center of pressure to the rear. To move the center of gravity forward, a heavier nose cone is used or a weight, such as clay, is added to the nose of the rocket. To move the center of pressure to the back, fins may be made larger or moved farther back on the body tube. Calculating the ratio of length to diameter in a rocket is another method to determine the potential for stability in a design. An ideal ratio is 10 to 1 (a 1-inch diameter rocket with a length of 10 inches). The computer program, Estes ASTROCAD : Performance Analysis for Model Rocketry, allows students to give data about the length, diameter and the location of CG of their rocket to predict the rocket s stability. VOCABULARY Stability: The tendency of a rocket with the proper center of gravity/center of pressure relationship to maintain a straight course despite rotating forces caused by variations in design and outside disturbances. STRATEGY Materials needed for each student: A completed model rocket, cardboard, ruler, string, tape, worksheet to record data about individual rocket and worksheet to evaluate the stability of each student s rocket and what needs to be done for the rocket to achieve stability. ESTES 22 EDUCATOR

MOTIVATION: Use the overhead transparencies (stability, CG and CP, stable and unstable flight) to show the relationship between CG and CP in rocket stability. Use the information in the background section of this unit to make the concept clear. A swing test or wind tunnel test using a rocket constructed by the teacher is an excellent way to demonstrate rocket stability. A. As the students ready their rockets for the swing test, they can observe the relationship of the center of gravity, which is where the string will be, and the lateral center of pressure of the rocket. For students who understand the equations for calculating the CP: B. If the computer program is to be used to predict stability of the rocket, the students should enter the data about their rocket on the data sheet. This program calculates the CP and gives the CG for student rockets. The calculations for CP use the same equations that the students used in the unit on center of pressure. C. Allow each student to do the swing test on their rocket, complete the evaluation sheet and make any alterations necessary. Evaluation: Observation of each student s swing test and evaluation of worksheets. NOTES ESTES 23 EDUCATOR

Lesson 5 (2 to 3 days) Math and Rocket Flight - Launching a Rocket Objectives of the Lesson: The student will be able to: Participate appropriately in the launching of each student s rocket. Demonstrate proper safety procedures during a launch. Record flight data on a class chart and on an individual chart. Track the flight of model rockets using an altitude measuring device to determine the angular distance the rocket traveled from launch to apogee. Use mathematical equations to determine the altitude or height reached by the model rocket flight using the data collected. Graphically determine the altitude reached by the model rocket. Use mathematical equations to determine velocities and accelerations during the rocket flight and describe their relationship to Newton s Second Law. BACKGROUND FOR THE TEACHER The launch area should be large enough, clear of people and clear of any easy to burn materials. On the day of launch, the wind speed should not be more than 20 mph. Early morning or early evening when there is little wind is the best time of the day to launch model rockets. The launch pad and the launch wire should be anchored down by bricks or something similar. The safety cap should be on the launch rod at all times except during the launch. The teacher should be in possession of the safety key at all times. DETERMINING ALTITUDE: Students will be interested in how high their rockets went. Accurate determination of heights reached requires care and precision in measuring, recording and calculating. Tracking: First, determine the length of the baseline. The baseline is the distance between the launcher and the observer or tracker with an altitude measuring device. Accuracy in measuring the baseline is very important in determining altitude. The baseline must be level with the tracking stations on the same level with the launcher. Next, determine the angular distance the rocket travels from launch to apogee (maximum altitude). The angular distance is determined by measuring the change in elevation angle, as seen by the tracker, between the rocket s position on the launch pad and apogee reached by the rocket in flight. A measuring device, such as the Estes AltiTrak, is used to find angular distance. The use of the device involves tracking the rocket from the launch pad to apogee, noting and recording the angular distance and then determining the actual height reached by the rocket by the use of a mathematical formula or plotting the information on graph paper. Set up a tracking system that suits the needs of your group. Accuracy in making and recording all measurements is very important. The simplest is one station tracking. The results are generally reliable. In one station, there is one baseline and one observer using an altitude measuring device such as the Estes AltiTrak. One station tracking assumes that the flight will be almost vertical. ESTES 24 EDUCATOR

If only one elevation tracker is used, it is a good idea to station it at a right angle to the wind flow. For example, if the wind is blowing to the west, the tracker should be either north or south of the launcher (Be careful not to stare into the sun). In this way we will keep the angle at C as close to a right angle as possible. By experimenting with a protractor and a straight edge, the rocketeer can demonstrate why the error would be less if the tracker is on a line at a right angle to the flow of the wind. In the figure above, the wind is blowing from B to D. The rocket is launched at point C, weathercocks into the wind following approximately line CA and at its maximum altitude is at point A. If the tracker is downwind from the launcher, the rocket will be seen at point F, and compute the altitude as the distance from C to F. The computed altitudes will be considerably lower than the true altitudes. On the other hand, if the rocket drifts toward the tracker, the computed altitude will be considerably higher than the true altitude. However, if the tracker is at point X in the following figure and the launcher at Y, then the rocket will appear to be at point A. Although the distance from the tracker to point y 1 will be slightly greater than the baseline used in computing the altitude, the error will not be nearly as great. Also, the small additional distance will serve to make altitude readings more conservative, as the baseline is increased. ESTES 25 EDUCATOR

By observing the proper relation between wind direction and the position of the tracker, we can generally determine with 90% or better accuracy the altitude the rocket reaches from data given by only one elevation tracker. Thus, on a calm day with a good model, almost perfect accuracy can be approached. Students can make an altitude tracking device using a straw, a protractor, string and an eraser. (Worksheet in Appendix) CALCULATIONS One Station Tracking - Elementary and Early Middle School The formula for determining the height reached by a model rocket flight using one station tracking is: Height = Baseline x Tangent of Angular distance B angular distance 90 A baseline C The tangent is used to determine altitude because the tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. In this example, the adjacent side is the distance along the baseline. The opposite side is the distance from the launcher to the rocket s maximum altitude. Tangents can be found in the Table of Tangents in the Appendix. For younger students that may not be able to comprehend the idea of a tangent, a graphical approach can be used to calculate the altitude. To use this method, a student must be able to plot angles with a protractor and layout scaled distances on graph paper. Referring to the graph in the figure, use the horizontal axis to plot the baseline distance (distance from launcher to tracker). With the same scale, use the vertical axis to plot the rocket s altitude. The rocket is launched at the origin on the graph paper and climbs vertically up the vertical axis. Mark the tracker s position on the ESTES 26 EDUCATOR

horizontal axis (250 feet) and plot the elevation angle (62 ). Extend the angle (line of sight) until it intersects the vertical axis. The intersected point on the vertical axis is the rocket s altitude. (Blank graph paper for plotting actual flight in Appendix). 470 500 400 Altitude (feet or meters) 300 200 100 62 A 0 100 200 300 400 500 Baseline distance (feet or meters) If we assume that the rocket flight is vertical, we can call angle C (in the previous figure) a right angle, 90. B is equal to 90 minus A because the sum of the angles in a triangle is 180. By definition: tan = opp adj so to find the distance from C to B or the height the rocket reached, take the tangent of Angle A times the distance along the baseline, side AC. Example: Baseline = 250 ft. Angle observed by tracker = 62 Tangent of 62 = 1.88 H = 250 ft. x 1.88 H = 470 ft. Two Station Tracking - Middle School and High School: A. When using two trackers without azimuth readings, the tracking stations are set up on opposite sides of the launcher. Preferably, to obtain the greatest accuracy, the stations should be in line with the wind, unlike the system used in single station tracking. If the wind is blowing to the south, one station will be north and the other south of the launch area. B. In the simplest method of two station tracking, the angles from each station are used together to get one value of height. C. This more accurate system of two station tracking uses two tracking stations placed on opposite sides of the launch pad in line with the wind. It uses sines instead of tangents. The formula for this method is: CD = tana = BC AC c sin A sin B where C = 180 ο ( A+ B) sinc For example, stations A and B are located on a 1000 ft. baseline with the launcher between them. Station A calls in an angular distance of 34 and station B calls in an angular distance of 22. The total of these two angles is 56. Therefore angle C, located ESTES 27 EDUCATOR